On the rank of the elliptic curves with a rational point of order 6 (Q883653)

The author is interested in the existence of elliptic curves over \(\mathbb Q\) of high rank, with a rational point of order \(6\). O. Lecacheux has previously obtained an elliptic curve over \(\mathbb Q(t)\) whose rank is at least \(3\) with a rational point of order \(6\). In his paper, the author constructs an infinite family of elliptic curves over \(\mathbb Q\), of rank at least \(4\) over \(\mathbb Q\), with a rational point of order \(6\). This family is parametrized by the rational points of the elliptic curve given by the equation \(y^2=x(x+4)(x+137)\), whose rank over \(\mathbb Q\) is equal to \(1\). This latest assertion can be justified by verifying that the derivative at \(1\) of its \(L\) function is not zero: it is equal about to \(7,8034\).